Question: Factor the following expression: $-5$ $x^2$ $-12$ $x$ $-7$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(-7)} &=& 35 \\ {a} + {b} &=& & & {-12} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $35$ and add them together. The factors that add up to ${-12}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${-5}$ $ \begin{eqnarray} {ab} &=& ({-7})({-5}) &=& 35 \\ {a} + {b} &=& {-7} + {-5} &=& -12 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-5}x^2 {-7}x {-5}x {-7} $ Group the terms so that there is a common factor in each group: $ ({-5}x^2 {-7}x) + ({-5}x {-7}) $ Factor out the common factors: $ x(-5x - 7) + 1(-5x - 7) $ Notice how $(-5x - 7)$ has become a common factor. Factor this out to find the answer. $(-5x - 7)(x + 1)$